Sample quiz on equations of altitudes
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1. In a $\triangle ABC$, an altitude from vertex $A$ is $\cdots$?
   
   a line from $A$ that bisects side $BC$ at $45^{\circ}$
   a line from $A$ that meets side $BC$ at $90^{\circ}$
   a line from $A$ that bisects side $BC$ at $90^{\circ}$
   a line from $A$ to the midpoint of side $BC$.
   
2. The point of intersection of the three altitudes of a triangle is known as $\cdots$?
   
   centroid
   circumcenter
   orthocenter
   altocenter
3. If a triangle contains an obtuse angle, how many of its altitudes are external (outside the triangle)?
   
   $0$
   $1$
   $2$
   $3$
4. Given $\triangle ABC$ with vertices at $A(1,2),~B(2,1),~C(3,3)$, find the equation of the altitude from $A$.
   
   $y=x$
   $y=-x$
   $y=-\frac{1}{2}x+\frac{5}{2}$
   $y=-2x+5$
5. Given $\triangle ABC$ with vertices at $A(1,2),~B(2,1),~C(3,3)$, find the equation of the altitude from $B$.
   
   $y=x$
   $y=-x$
   $y=-\frac{1}{2}x+\frac{5}{2}$
   $y=-2x+5$
6. Given $\triangle ABC$ with vertices at $A(1,2),~B(2,1),~C(3,3)$, find the equation of the altitude from $C$.
   
   $y=x$
   $y=-x$
   $y=-\frac{1}{2}x+\frac{5}{2}$
   $y=-2x+5$
7. Given $\triangle ABC$ with vertices at $A(1,2),~B(2,1),~C(3,3)$, find the (coordinates of the) foot of the altitude from $C$.
   
   $(\frac{3}{2},\frac{3}{2})$
   $(\frac{2}{3},\frac{3}{2})$
   $(\frac{3}{2},\frac{2}{3})$
   $(\frac{2}{3},\frac{2}{3})$
8. Find the orthocenter of $\triangle ABC$ whose vertices are located at $A(1,2),~(2,1),~C(3,3)$.
   
   $(\frac{3}{5},\frac{3}{5})$
   $(\frac{5}{3},\frac{5}{3})$
   $(-\frac{5}{3},\frac{5}{3})$
   $(-\frac{3}{5},\frac{3}{5})$
9. Given $\triangle ABC$ with vertices at $A(0,0),~B(1,4),~C(3,6)$, find the equation of the altitude from $A$.
   
   $y=-x$
   $y=-x+1$
   $y=-\frac{1}{2}x+\frac{9}{2}$
   $y=-\frac{1}{4}x+\frac{27}{4}$
10. Given $\triangle ABC$ with vertices at $A(0,0),~B(1,4),~C(3,6)$, find the equation of the altitude from $B$.
    
    $y=-x$
    $y=-x+1$
    $y=-\frac{1}{2}x+\frac{9}{2}$
    $y=-\frac{1}{4}x+\frac{27}{4}$
11. Given $\triangle ABC$ with vertices at $A(0,0),~B(1,4),~C(3,6)$, find the equation of the altitude from $C$.
    
    $y=-x$
    $y=-x+1$
    $y=-\frac{1}{2}x+\frac{9}{2}$
    $y=-\frac{1}{4}x+\frac{27}{4}$
12. Find the orthocenter of $\triangle ABC$ with vertices at $A(0,0),~B(1,4),~C(3,6)$.
    
    $(-9,9)$
    $(9,-9)$
    $(9,9)$
    $(0,9)$
13. Consider $\triangle ABC$ in which $\angle C=90^{\circ}$. This triangle's orthocenter is located at $\cdots$?
    
    $A$
    $B$
    $C$
    $AB$
14. Find the distance between the centroid and the foot of the altitude from $C$, given $\triangle ABC$ with vertices at $A(1,2),~B(2,1),~C(3,3)$.
    
    $1$
    $2$
    $\sqrt{2}$
    $\frac{\sqrt{2}}{2}$
15. Given $\triangle ABC$ with vertices at $A(0,0),~B(1,4),~C(3,6)$, find the LENGTH of the altitude from $A$.
    
    $\frac{3}{2}$
    $\frac{2}{3}$
    $\frac{3}{2}\sqrt{2}$
    $\frac{2}{3}\sqrt{2}$